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Simplifying 0 = 2a2 + -5a + -1 Reorder the terms: 0 = -1 + -5a + 2a2 Solving 0 = -1 + -5a + 2a2 Solving for variable 'a'. Combine like terms: 0 + 1 = 1 1 + 5a + -2a2 = -1 + -5a + 2a2 + 1 + 5a + -2a2 Reorder the terms: 1 + 5a + -2a2 = -1 + 1 + -5a + 5a + 2a2 + -2a2 Combine like terms: -1 + 1 = 0 1 + 5a + -2a2 = 0 + -5a + 5a + 2a2 + -2a2 1 + 5a + -2a2 = -5a + 5a + 2a2 + -2a2 Combine like terms: -5a + 5a = 0 1 + 5a + -2a2 = 0 + 2a2 + -2a2 1 + 5a + -2a2 = 2a2 + -2a2 Combine like terms: 2a2 + -2a2 = 0 1 + 5a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -2.5a + a2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -2.5a + 0.5 + a2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -2.5a + a2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -2.5a + a2 = 0 + 0.5 -2.5a + a2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -2.5a + a2 = 0.5 The a term is -2.5a. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5a + 1.5625 + a2 = 0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5a + a2 = 0.5 + 1.5625 Combine like terms: 0.5 + 1.5625 = 2.0625 1.5625 + -2.5a + a2 = 2.0625 Factor a perfect square on the left side: (a + -1.25)(a + -1.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (a + -1.25) equal to 1.436140662 and -1.436140662.Subproblem 1
a + -1.25 = 1.436140662 Simplifying a + -1.25 = 1.436140662 Reorder the terms: -1.25 + a = 1.436140662 Solving -1.25 + a = 1.436140662 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = 1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = 1.436140662 + 1.25 a = 1.436140662 + 1.25 Combine like terms: 1.436140662 + 1.25 = 2.686140662 a = 2.686140662 Simplifying a = 2.686140662Subproblem 2
a + -1.25 = -1.436140662 Simplifying a + -1.25 = -1.436140662 Reorder the terms: -1.25 + a = -1.436140662 Solving -1.25 + a = -1.436140662 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = -1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = -1.436140662 + 1.25 a = -1.436140662 + 1.25 Combine like terms: -1.436140662 + 1.25 = -0.186140662 a = -0.186140662 Simplifying a = -0.186140662Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.686140662, -0.186140662}
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